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<li class="navelem"><a class="el" href="dir_296d53ceaeaa7e099814a6def439fe8a.html">math</a></li>  </ul>
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<div class="title">useful_math.cpp</div>  </div>
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<div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno">    1</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l00002"></a><span class="lineno">    2</span>&#160;<span class="comment">This is part of OpenLoong Dynamics Control, an open project for the control of biped robot,</span></div>
<div class="line"><a name="l00003"></a><span class="lineno">    3</span>&#160;<span class="comment">Copyright (C) 2024 Humanoid Robot (Shanghai) Co., Ltd, under Apache 2.0.</span></div>
<div class="line"><a name="l00004"></a><span class="lineno">    4</span>&#160;<span class="comment">Feel free to use in any purpose, and cite OpenLoong-Dynamics-Control in any style, to contribute to the advancement of the community.</span></div>
<div class="line"><a name="l00005"></a><span class="lineno">    5</span>&#160;<span class="comment"> &lt;https://atomgit.com/openloong/openloong-dyn-control.git&gt;</span></div>
<div class="line"><a name="l00006"></a><span class="lineno">    6</span>&#160;<span class="comment"> &lt;web@openloong.org.cn&gt;</span></div>
<div class="line"><a name="l00007"></a><span class="lineno">    7</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00008"></a><span class="lineno">    8</span>&#160;<span class="preprocessor">#include &quot;useful_math.h&quot;</span></div>
<div class="line"><a name="l00009"></a><span class="lineno">    9</span>&#160;<span class="preprocessor">#include &quot;Eigen/Dense&quot;</span></div>
<div class="line"><a name="l00010"></a><span class="lineno">   10</span>&#160; </div>
<div class="line"><a name="l00011"></a><span class="lineno">   11</span>&#160;<span class="keyword">using namespace </span>Eigen;</div>
<div class="line"><a name="l00012"></a><span class="lineno">   12</span>&#160; </div>
<div class="line"><a name="l00013"></a><span class="lineno">   13</span>&#160; </div>
<div class="line"><a name="l00014"></a><span class="lineno">   14</span>&#160;<span class="comment">// SVD based pseudo-inverse</span></div>
<div class="line"><a name="l00015"></a><span class="lineno">   15</span>&#160;Eigen::MatrixXd pseudoInv_SVD(<span class="keyword">const</span> Eigen::MatrixXd &amp;mat)</div>
<div class="line"><a name="l00016"></a><span class="lineno">   16</span>&#160;{</div>
<div class="line"><a name="l00017"></a><span class="lineno">   17</span>&#160;    <span class="keyword">typedef</span> <span class="keyword">typename</span> Eigen::MatrixXd::Scalar Scalar;</div>
<div class="line"><a name="l00018"></a><span class="lineno">   18</span>&#160;    <span class="keyword">auto</span> svd = mat.jacobiSvd(Eigen::ComputeFullU | Eigen::ComputeFullV);</div>
<div class="line"><a name="l00019"></a><span class="lineno">   19</span>&#160;    <span class="keyword">const</span> <span class="keyword">auto</span> &amp;singularValues = svd.singularValues();</div>
<div class="line"><a name="l00020"></a><span class="lineno">   20</span>&#160;    Eigen::MatrixXd singularValuesInv;</div>
<div class="line"><a name="l00021"></a><span class="lineno">   21</span>&#160;    <span class="keywordtype">double</span> maxSgVal{0};</div>
<div class="line"><a name="l00022"></a><span class="lineno">   22</span>&#160;    <span class="keywordflow">for</span> (<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> i = 0; i &lt; singularValues.size(); ++i) {</div>
<div class="line"><a name="l00023"></a><span class="lineno">   23</span>&#160;        <span class="keywordflow">if</span> (singularValues(i) &gt; maxSgVal)</div>
<div class="line"><a name="l00024"></a><span class="lineno">   24</span>&#160;            maxSgVal=singularValues(i);</div>
<div class="line"><a name="l00025"></a><span class="lineno">   25</span>&#160;    }</div>
<div class="line"><a name="l00026"></a><span class="lineno">   26</span>&#160;    <span class="keywordtype">double</span> tolerance= std::numeric_limits&lt;double&gt;::epsilon() * std::max(mat.rows(), mat.cols()) * maxSgVal;</div>
<div class="line"><a name="l00027"></a><span class="lineno">   27</span>&#160;    singularValuesInv=Eigen::MatrixXd::Zero(mat.cols(),mat.rows());</div>
<div class="line"><a name="l00028"></a><span class="lineno">   28</span>&#160;    <span class="keywordflow">for</span> (<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> i = 0; i &lt; singularValues.size(); ++i) {</div>
<div class="line"><a name="l00029"></a><span class="lineno">   29</span>&#160;        <span class="keywordflow">if</span> (singularValues(i) &gt; tolerance) {</div>
<div class="line"><a name="l00030"></a><span class="lineno">   30</span>&#160;            singularValuesInv(i, i) = Scalar{1} / singularValues(i);</div>
<div class="line"><a name="l00031"></a><span class="lineno">   31</span>&#160;        } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00032"></a><span class="lineno">   32</span>&#160;            singularValuesInv(i, i) = Scalar{0};</div>
<div class="line"><a name="l00033"></a><span class="lineno">   33</span>&#160;        }</div>
<div class="line"><a name="l00034"></a><span class="lineno">   34</span>&#160;    }</div>
<div class="line"><a name="l00035"></a><span class="lineno">   35</span>&#160;    <span class="keywordflow">return</span> svd.matrixV() * singularValuesInv * svd.matrixU().adjoint();</div>
<div class="line"><a name="l00036"></a><span class="lineno">   36</span>&#160;};</div>
<div class="line"><a name="l00037"></a><span class="lineno">   37</span>&#160; </div>
<div class="line"><a name="l00038"></a><span class="lineno">   38</span>&#160;Eigen::MatrixXd pseudoInv_right(<span class="keyword">const</span> Eigen::MatrixXd &amp;M) {</div>
<div class="line"><a name="l00039"></a><span class="lineno">   39</span>&#160;    Eigen::MatrixXd Mres;</div>
<div class="line"><a name="l00040"></a><span class="lineno">   40</span>&#160;    Mres=M*M.transpose();</div>
<div class="line"><a name="l00041"></a><span class="lineno">   41</span>&#160;    Mres=M.transpose()* Mres.ldlt().solve(Eigen::MatrixXd::Identity(Mres.rows(), Mres.cols()));</div>
<div class="line"><a name="l00042"></a><span class="lineno">   42</span>&#160;<span class="comment">//    Mres=M.transpose()* Mres.inverse();</span></div>
<div class="line"><a name="l00043"></a><span class="lineno">   43</span>&#160;    <span class="keywordflow">return</span> Mres;</div>
<div class="line"><a name="l00044"></a><span class="lineno">   44</span>&#160;}</div>
<div class="line"><a name="l00045"></a><span class="lineno">   45</span>&#160; </div>
<div class="line"><a name="l00046"></a><span class="lineno">   46</span>&#160;Eigen::MatrixXd pseudoInv_right_weighted(<span class="keyword">const</span> Eigen::MatrixXd &amp;M, <span class="keyword">const</span> Eigen::DiagonalMatrix&lt;double, -1&gt; &amp;W) {</div>
<div class="line"><a name="l00047"></a><span class="lineno">   47</span>&#160;    <span class="keywordtype">double</span> damp=0;</div>
<div class="line"><a name="l00048"></a><span class="lineno">   48</span>&#160;    Eigen::MatrixXd Mres;</div>
<div class="line"><a name="l00049"></a><span class="lineno">   49</span>&#160;    Mres=M*W.inverse()*M.transpose();</div>
<div class="line"><a name="l00050"></a><span class="lineno">   50</span>&#160;<span class="comment">//    Mres=W.inverse()*M.transpose()* pseudoInv_SVD(Mres);</span></div>
<div class="line"><a name="l00051"></a><span class="lineno">   51</span>&#160;    Mres.diagonal().array() += damp;</div>
<div class="line"><a name="l00052"></a><span class="lineno">   52</span>&#160;    Mres=W.inverse()*M.transpose()* Mres.completeOrthogonalDecomposition().pseudoInverse();</div>
<div class="line"><a name="l00053"></a><span class="lineno">   53</span>&#160;    <span class="keywordflow">return</span> Mres;</div>
<div class="line"><a name="l00054"></a><span class="lineno">   54</span>&#160;}</div>
<div class="line"><a name="l00055"></a><span class="lineno">   55</span>&#160; </div>
<div class="line"><a name="l00056"></a><span class="lineno">   56</span>&#160;<span class="comment">//Eigen::MatrixXd dyn_pseudoInv(const MatrixXd &amp;M, const MatrixXd &amp;dyn_A) {</span></div>
<div class="line"><a name="l00058"></a><span class="lineno">   58</span>&#160;<span class="comment"></span><span class="comment">//    Eigen::MatrixXd res;</span></div>
<div class="line"><a name="l00059"></a><span class="lineno">   59</span>&#160;<span class="comment">//    Eigen::MatrixXd Minv = dyn_A.ldlt().solve(Eigen::MatrixXd::Identity(dyn_A.rows(), dyn_A.cols()));</span></div>
<div class="line"><a name="l00061"></a><span class="lineno">   61</span>&#160;<span class="comment"></span><span class="comment">//    res = Minv * M.transpose() * (M * Minv * M.transpose()).inverse();</span></div>
<div class="line"><a name="l00062"></a><span class="lineno">   62</span>&#160;<span class="comment">//    return res;</span></div>
<div class="line"><a name="l00063"></a><span class="lineno">   63</span>&#160;<span class="comment">//}</span></div>
<div class="line"><a name="l00064"></a><span class="lineno">   64</span>&#160; </div>
<div class="line"><a name="l00065"></a><span class="lineno">   65</span>&#160;Eigen::MatrixXd dyn_pseudoInv(<span class="keyword">const</span> Eigen::MatrixXd &amp;M, <span class="keyword">const</span> Eigen::MatrixXd &amp;dyn_M, <span class="keywordtype">bool</span> isMinv) {</div>
<div class="line"><a name="l00066"></a><span class="lineno">   66</span>&#160;    <span class="keywordtype">double</span> damp=0;</div>
<div class="line"><a name="l00067"></a><span class="lineno">   67</span>&#160;    Eigen::MatrixXd Minv;</div>
<div class="line"><a name="l00068"></a><span class="lineno">   68</span>&#160; </div>
<div class="line"><a name="l00069"></a><span class="lineno">   69</span>&#160;    <span class="keywordflow">if</span> (isMinv)</div>
<div class="line"><a name="l00070"></a><span class="lineno">   70</span>&#160;        Minv = dyn_M;</div>
<div class="line"><a name="l00071"></a><span class="lineno">   71</span>&#160;    <span class="keywordflow">else</span></div>
<div class="line"><a name="l00072"></a><span class="lineno">   72</span>&#160;        Minv = dyn_M.llt().solve(Eigen::MatrixXd::Identity(dyn_M.rows(), dyn_M.cols()));</div>
<div class="line"><a name="l00073"></a><span class="lineno">   73</span>&#160; </div>
<div class="line"><a name="l00074"></a><span class="lineno">   74</span>&#160;    Eigen::MatrixXd temp = M * Minv * M.transpose();</div>
<div class="line"><a name="l00075"></a><span class="lineno">   75</span>&#160; </div>
<div class="line"><a name="l00076"></a><span class="lineno">   76</span>&#160;    temp.diagonal().array() += damp;</div>
<div class="line"><a name="l00077"></a><span class="lineno">   77</span>&#160; </div>
<div class="line"><a name="l00078"></a><span class="lineno">   78</span>&#160;    Eigen::MatrixXd res = Minv * M.transpose() * temp.completeOrthogonalDecomposition().pseudoInverse();</div>
<div class="line"><a name="l00079"></a><span class="lineno">   79</span>&#160; </div>
<div class="line"><a name="l00080"></a><span class="lineno">   80</span>&#160;<span class="comment">//    Eigen::MatrixXd res = Minv * M.transpose() * temp.inverse();</span></div>
<div class="line"><a name="l00081"></a><span class="lineno">   81</span>&#160;    <span class="keywordflow">return</span> res;</div>
<div class="line"><a name="l00082"></a><span class="lineno">   82</span>&#160;}</div>
<div class="line"><a name="l00083"></a><span class="lineno">   83</span>&#160; </div>
<div class="line"><a name="l00084"></a><span class="lineno">   84</span>&#160;Eigen::Matrix&lt;double, 3, 3&gt; eul2Rot(<span class="keywordtype">double</span> roll, <span class="keywordtype">double</span> pitch, <span class="keywordtype">double</span> yaw) {</div>
<div class="line"><a name="l00085"></a><span class="lineno">   85</span>&#160;    Eigen::Matrix&lt;double, 3, 3&gt; Rx, Ry, Rz;</div>
<div class="line"><a name="l00086"></a><span class="lineno">   86</span>&#160;    Rz &lt;&lt; cos(yaw), -sin(yaw), 0,</div>
<div class="line"><a name="l00087"></a><span class="lineno">   87</span>&#160;            sin(yaw), cos(yaw), 0,</div>
<div class="line"><a name="l00088"></a><span class="lineno">   88</span>&#160;            0, 0, 1;</div>
<div class="line"><a name="l00089"></a><span class="lineno">   89</span>&#160;    Ry &lt;&lt; cos(pitch), 0, sin(pitch),</div>
<div class="line"><a name="l00090"></a><span class="lineno">   90</span>&#160;            0, 1, 0,</div>
<div class="line"><a name="l00091"></a><span class="lineno">   91</span>&#160;            -sin(pitch), 0, cos(pitch);</div>
<div class="line"><a name="l00092"></a><span class="lineno">   92</span>&#160;    Rx &lt;&lt; 1, 0, 0,</div>
<div class="line"><a name="l00093"></a><span class="lineno">   93</span>&#160;            0, cos(roll), -sin(roll),</div>
<div class="line"><a name="l00094"></a><span class="lineno">   94</span>&#160;            0, sin(roll), cos(roll);</div>
<div class="line"><a name="l00095"></a><span class="lineno">   95</span>&#160;    <span class="keywordflow">return</span> Rz * Ry * Rx;</div>
<div class="line"><a name="l00096"></a><span class="lineno">   96</span>&#160;}</div>
<div class="line"><a name="l00097"></a><span class="lineno">   97</span>&#160; </div>
<div class="line"><a name="l00098"></a><span class="lineno">   98</span>&#160;Eigen::Vector3d Rot2eul(<span class="keyword">const</span> Eigen::Matrix3d &amp;rot){</div>
<div class="line"><a name="l00099"></a><span class="lineno">   99</span>&#160;    Eigen::Vector3d eul;</div>
<div class="line"><a name="l00100"></a><span class="lineno">  100</span>&#160;    eul(0) = std::atan2(rot(2, 1), rot(2, 2));</div>
<div class="line"><a name="l00101"></a><span class="lineno">  101</span>&#160;    eul(1) = std::atan2(-rot(2, 0), std::sqrt(rot(2, 1) * rot(2, 1) + rot(2, 2) * rot(2, 2)));</div>
<div class="line"><a name="l00102"></a><span class="lineno">  102</span>&#160;    eul(2) = std::atan2(rot(1, 0), rot(0, 0));</div>
<div class="line"><a name="l00103"></a><span class="lineno">  103</span>&#160;    <span class="keywordflow">return</span> eul;</div>
<div class="line"><a name="l00104"></a><span class="lineno">  104</span>&#160;}</div>
<div class="line"><a name="l00105"></a><span class="lineno">  105</span>&#160; </div>
<div class="line"><a name="l00106"></a><span class="lineno">  106</span>&#160;std::vector&lt;double&gt; eigen2std(<span class="keyword">const</span> Eigen::VectorXd &amp;eigenVector){</div>
<div class="line"><a name="l00107"></a><span class="lineno">  107</span>&#160;    std::vector&lt;double&gt; stdVector(eigenVector.data(), eigenVector.data() + eigenVector.size());</div>
<div class="line"><a name="l00108"></a><span class="lineno">  108</span>&#160;    <span class="keywordflow">return</span> stdVector;</div>
<div class="line"><a name="l00109"></a><span class="lineno">  109</span>&#160;}</div>
<div class="line"><a name="l00110"></a><span class="lineno">  110</span>&#160; </div>
<div class="line"><a name="l00111"></a><span class="lineno">  111</span>&#160;Eigen::Quaterniond eul2quat(<span class="keywordtype">double</span> roll, <span class="keywordtype">double</span> pitch, <span class="keywordtype">double</span> yaw) {</div>
<div class="line"><a name="l00112"></a><span class="lineno">  112</span>&#160;    Eigen::Matrix3d R = eul2Rot(roll, pitch, yaw);</div>
<div class="line"><a name="l00113"></a><span class="lineno">  113</span>&#160;    Eigen::Quaternion&lt;double&gt; quatCur;</div>
<div class="line"><a name="l00114"></a><span class="lineno">  114</span>&#160;    quatCur = R; <span class="comment">//rotation matrix converted to quaternion</span></div>
<div class="line"><a name="l00115"></a><span class="lineno">  115</span>&#160;    Eigen::Quaterniond resQuat;</div>
<div class="line"><a name="l00116"></a><span class="lineno">  116</span>&#160;    resQuat = quatCur;</div>
<div class="line"><a name="l00117"></a><span class="lineno">  117</span>&#160;    <span class="keywordflow">return</span> resQuat;</div>
<div class="line"><a name="l00118"></a><span class="lineno">  118</span>&#160;}</div>
<div class="line"><a name="l00119"></a><span class="lineno">  119</span>&#160; </div>
<div class="line"><a name="l00120"></a><span class="lineno">  120</span>&#160;Eigen::Matrix&lt;double, 3, 1&gt; diffRot(<span class="keyword">const</span> Eigen::Matrix3d &amp;Rcur, Eigen::Matrix3d &amp;Rdes) {</div>
<div class="line"><a name="l00121"></a><span class="lineno">  121</span>&#160;    Eigen::Matrix3d R = Rcur.transpose() * Rdes;</div>
<div class="line"><a name="l00122"></a><span class="lineno">  122</span>&#160;    Eigen::Vector3d w;</div>
<div class="line"><a name="l00123"></a><span class="lineno">  123</span>&#160; </div>
<div class="line"><a name="l00124"></a><span class="lineno">  124</span>&#160;    <span class="keywordflow">if</span> (R.isDiagonal(1e-5) &amp;&amp; fabs(R(0, 0)) + fabs(R(1, 1)) + fabs(R(2, 2)) - 3 &lt; 1e-3) {</div>
<div class="line"><a name="l00125"></a><span class="lineno">  125</span>&#160;        w.setZero();</div>
<div class="line"><a name="l00126"></a><span class="lineno">  126</span>&#160;    } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (R.isDiagonal(1e-5)) {</div>
<div class="line"><a name="l00127"></a><span class="lineno">  127</span>&#160;        w &lt;&lt; R(0, 0) + 1, R(1, 1) + 1, R(2, 2) + 1;</div>
<div class="line"><a name="l00128"></a><span class="lineno">  128</span>&#160;        w = w * 3.1415 / 2.0;</div>
<div class="line"><a name="l00129"></a><span class="lineno">  129</span>&#160;    } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00130"></a><span class="lineno">  130</span>&#160;        Eigen::Vector3d l;</div>
<div class="line"><a name="l00131"></a><span class="lineno">  131</span>&#160;        l &lt;&lt; R(2, 1) - R(1, 2), R(0, 2) - R(2, 0), R(1, 0) - R(0, 1);</div>
<div class="line"><a name="l00132"></a><span class="lineno">  132</span>&#160;        <span class="keywordtype">double</span> sita = atan2(l.norm(), R(0, 0) + R(1, 1) + R(2, 2) - 1);</div>
<div class="line"><a name="l00133"></a><span class="lineno">  133</span>&#160;        w = sita * l / l.norm();</div>
<div class="line"><a name="l00134"></a><span class="lineno">  134</span>&#160;    }</div>
<div class="line"><a name="l00135"></a><span class="lineno">  135</span>&#160;    w = Rcur * w;</div>
<div class="line"><a name="l00136"></a><span class="lineno">  136</span>&#160;    <span class="keywordflow">return</span> w;</div>
<div class="line"><a name="l00137"></a><span class="lineno">  137</span>&#160;}</div>
<div class="line"><a name="l00138"></a><span class="lineno">  138</span>&#160; </div>
<div class="line"><a name="l00139"></a><span class="lineno">  139</span>&#160;Eigen::Matrix&lt;double, 4, 1&gt; quat2axisAngle(<span class="keyword">const</span> Eigen::Quaternion&lt;double&gt; &amp;quat) {</div>
<div class="line"><a name="l00140"></a><span class="lineno">  140</span>&#160;    <span class="comment">// quat must be normolized</span></div>
<div class="line"><a name="l00141"></a><span class="lineno">  141</span>&#160;    Eigen::Matrix&lt;double, 4, 1&gt; res;</div>
<div class="line"><a name="l00142"></a><span class="lineno">  142</span>&#160;    <span class="keywordtype">double</span> angle = 2.0 * acos(quat.w());</div>
<div class="line"><a name="l00143"></a><span class="lineno">  143</span>&#160;    <span class="keywordtype">double</span> s = sqrt(1 - quat.w() * quat.w());</div>
<div class="line"><a name="l00144"></a><span class="lineno">  144</span>&#160;    <span class="keywordflow">if</span> (s &lt; 1e-8) {</div>
<div class="line"><a name="l00145"></a><span class="lineno">  145</span>&#160;        res(0) = quat.x();</div>
<div class="line"><a name="l00146"></a><span class="lineno">  146</span>&#160;        res(1) = quat.y();</div>
<div class="line"><a name="l00147"></a><span class="lineno">  147</span>&#160;<span class="comment">//        res(2)=quat.z();</span></div>
<div class="line"><a name="l00148"></a><span class="lineno">  148</span>&#160;        res(2) = 1; <span class="comment">// no rotation now but the axis must not be [0,0,0]</span></div>
<div class="line"><a name="l00149"></a><span class="lineno">  149</span>&#160;    } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00150"></a><span class="lineno">  150</span>&#160;        res(0) = quat.x() / s;</div>
<div class="line"><a name="l00151"></a><span class="lineno">  151</span>&#160;        res(1) = quat.y() / s;</div>
<div class="line"><a name="l00152"></a><span class="lineno">  152</span>&#160;        res(2) = quat.z() / s;</div>
<div class="line"><a name="l00153"></a><span class="lineno">  153</span>&#160;    }</div>
<div class="line"><a name="l00154"></a><span class="lineno">  154</span>&#160;    res(3) = angle;</div>
<div class="line"><a name="l00155"></a><span class="lineno">  155</span>&#160;    <span class="keywordflow">return</span> res;</div>
<div class="line"><a name="l00156"></a><span class="lineno">  156</span>&#160;}</div>
<div class="line"><a name="l00157"></a><span class="lineno">  157</span>&#160; </div>
<div class="line"><a name="l00158"></a><span class="lineno">  158</span>&#160;Eigen::Quaterniond intQuat(<span class="keyword">const</span> Eigen::Quaterniond &amp;quat, <span class="keyword">const</span> Eigen::Matrix&lt;double, 3, 1&gt; &amp;w) {</div>
<div class="line"><a name="l00159"></a><span class="lineno">  159</span>&#160;    Eigen::Matrix3d Rcur = quat.normalized().toRotationMatrix();</div>
<div class="line"><a name="l00160"></a><span class="lineno">  160</span>&#160;    Eigen::Matrix3d Rinc = Eigen::Matrix3d::Identity();</div>
<div class="line"><a name="l00161"></a><span class="lineno">  161</span>&#160;    <span class="keywordtype">double</span> theta = w.norm();</div>
<div class="line"><a name="l00162"></a><span class="lineno">  162</span>&#160;    <span class="keywordflow">if</span> (theta &gt; 1e-4) {</div>
<div class="line"><a name="l00163"></a><span class="lineno">  163</span>&#160;        Eigen::Vector3d w_norm;</div>
<div class="line"><a name="l00164"></a><span class="lineno">  164</span>&#160;        w_norm = w / theta;</div>
<div class="line"><a name="l00165"></a><span class="lineno">  165</span>&#160;        Eigen::Matrix3d a;</div>
<div class="line"><a name="l00166"></a><span class="lineno">  166</span>&#160;        a &lt;&lt; 0, -w_norm(2), w_norm(1),</div>
<div class="line"><a name="l00167"></a><span class="lineno">  167</span>&#160;                w_norm(0), 0, -w_norm(0),</div>
<div class="line"><a name="l00168"></a><span class="lineno">  168</span>&#160;                -w_norm(1), w_norm(0), 0;</div>
<div class="line"><a name="l00169"></a><span class="lineno">  169</span>&#160;        Rinc = Eigen::Matrix3d::Identity() + a * sin(theta) + a * a * (1 - cos(theta));</div>
<div class="line"><a name="l00170"></a><span class="lineno">  170</span>&#160;    }</div>
<div class="line"><a name="l00171"></a><span class="lineno">  171</span>&#160;    Eigen::Matrix3d Rend = Rcur * Rinc;</div>
<div class="line"><a name="l00172"></a><span class="lineno">  172</span>&#160;    Eigen::Quaterniond quatRes;</div>
<div class="line"><a name="l00173"></a><span class="lineno">  173</span>&#160;    quatRes = Rend;</div>
<div class="line"><a name="l00174"></a><span class="lineno">  174</span>&#160;    <span class="keywordflow">return</span> quatRes;</div>
<div class="line"><a name="l00175"></a><span class="lineno">  175</span>&#160;}</div>
<div class="line"><a name="l00176"></a><span class="lineno">  176</span>&#160; </div>
<div class="line"><a name="l00177"></a><span class="lineno">  177</span>&#160;<span class="keywordtype">double</span> sign(<span class="keyword">const</span> <span class="keywordtype">double</span> &amp;x) {</div>
<div class="line"><a name="l00178"></a><span class="lineno">  178</span>&#160;    <span class="keywordflow">if</span> (x &gt; 0)</div>
<div class="line"><a name="l00179"></a><span class="lineno">  179</span>&#160;        <span class="keywordflow">return</span> 1.0;</div>
<div class="line"><a name="l00180"></a><span class="lineno">  180</span>&#160;    <span class="keywordflow">else</span> <span class="keywordflow">if</span> (x &lt; 0)</div>
<div class="line"><a name="l00181"></a><span class="lineno">  181</span>&#160;        <span class="keywordflow">return</span> -1.0;</div>
<div class="line"><a name="l00182"></a><span class="lineno">  182</span>&#160;    <span class="keywordflow">else</span></div>
<div class="line"><a name="l00183"></a><span class="lineno">  183</span>&#160;        <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00184"></a><span class="lineno">  184</span>&#160;}</div>
<div class="line"><a name="l00185"></a><span class="lineno">  185</span>&#160; </div>
<div class="line"><a name="l00186"></a><span class="lineno">  186</span>&#160;Eigen::Matrix&lt;double,3,3&gt; Rx3(<span class="keywordtype">double</span> theta){ <span class="comment">//local to world</span></div>
<div class="line"><a name="l00187"></a><span class="lineno">  187</span>&#160;    <span class="comment">// for 2D-XY vector, rotation matrix along z axis</span></div>
<div class="line"><a name="l00188"></a><span class="lineno">  188</span>&#160;    Eigen::Matrix&lt;double,3,3&gt; M;</div>
<div class="line"><a name="l00189"></a><span class="lineno">  189</span>&#160;    M &lt;&lt; 1, 0, 0,</div>
<div class="line"><a name="l00190"></a><span class="lineno">  190</span>&#160;            0, cos(theta), -sin(theta),</div>
<div class="line"><a name="l00191"></a><span class="lineno">  191</span>&#160;            0, sin(theta),  cos(theta);</div>
<div class="line"><a name="l00192"></a><span class="lineno">  192</span>&#160;    <span class="keywordflow">return</span> M;</div>
<div class="line"><a name="l00193"></a><span class="lineno">  193</span>&#160;}</div>
<div class="line"><a name="l00194"></a><span class="lineno">  194</span>&#160; </div>
<div class="line"><a name="l00195"></a><span class="lineno">  195</span>&#160;Eigen::Matrix&lt;double,3,3&gt; Ry3(<span class="keywordtype">double</span> theta){ <span class="comment">//local to world</span></div>
<div class="line"><a name="l00196"></a><span class="lineno">  196</span>&#160;    <span class="comment">// for 2D-XY vector, rotation matrix along z axis</span></div>
<div class="line"><a name="l00197"></a><span class="lineno">  197</span>&#160;    Eigen::Matrix&lt;double,3,3&gt; M;</div>
<div class="line"><a name="l00198"></a><span class="lineno">  198</span>&#160;    M &lt;&lt; cos(theta), 0, sin(theta),</div>
<div class="line"><a name="l00199"></a><span class="lineno">  199</span>&#160;            0,             1,               0,</div>
<div class="line"><a name="l00200"></a><span class="lineno">  200</span>&#160;            -sin(theta), 0, cos(theta);</div>
<div class="line"><a name="l00201"></a><span class="lineno">  201</span>&#160;    <span class="keywordflow">return</span> M;</div>
<div class="line"><a name="l00202"></a><span class="lineno">  202</span>&#160;}</div>
<div class="line"><a name="l00203"></a><span class="lineno">  203</span>&#160; </div>
<div class="line"><a name="l00204"></a><span class="lineno">  204</span>&#160;Eigen::Matrix&lt;double,3,3&gt; Rz3(<span class="keywordtype">double</span> theta){ <span class="comment">//local to world</span></div>
<div class="line"><a name="l00205"></a><span class="lineno">  205</span>&#160;    <span class="comment">// for 2D-XY vector, rotation matrix along z axis</span></div>
<div class="line"><a name="l00206"></a><span class="lineno">  206</span>&#160;    Eigen::Matrix&lt;double,3,3&gt; M;</div>
<div class="line"><a name="l00207"></a><span class="lineno">  207</span>&#160;    M &lt;&lt; cos(theta), -sin(theta), 0,</div>
<div class="line"><a name="l00208"></a><span class="lineno">  208</span>&#160;            sin(theta), cos(theta),  0,</div>
<div class="line"><a name="l00209"></a><span class="lineno">  209</span>&#160;            0,             0,              1;</div>
<div class="line"><a name="l00210"></a><span class="lineno">  210</span>&#160;    <span class="keywordflow">return</span> M;</div>
<div class="line"><a name="l00211"></a><span class="lineno">  211</span>&#160;}</div>
<div class="line"><a name="l00212"></a><span class="lineno">  212</span>&#160; </div>
<div class="line"><a name="l00213"></a><span class="lineno">  213</span>&#160;Eigen::Matrix&lt;double,3,3&gt; CrossProduct_A(Eigen::Matrix&lt;double,3,1&gt;  A){</div>
<div class="line"><a name="l00214"></a><span class="lineno">  214</span>&#160;    Eigen::Matrix&lt;double,3,3&gt; M;</div>
<div class="line"><a name="l00215"></a><span class="lineno">  215</span>&#160;    M &lt;&lt; 0.0, -A[2], A[1],</div>
<div class="line"><a name="l00216"></a><span class="lineno">  216</span>&#160;            A[2], 0.0, -A[0],</div>
<div class="line"><a name="l00217"></a><span class="lineno">  217</span>&#160;            -A[1], A[0], 0.0;</div>
<div class="line"><a name="l00218"></a><span class="lineno">  218</span>&#160;    <span class="keywordflow">return</span> M;</div>
<div class="line"><a name="l00219"></a><span class="lineno">  219</span>&#160;}</div>
<div class="line"><a name="l00220"></a><span class="lineno">  220</span>&#160; </div>
<div class="line"><a name="l00221"></a><span class="lineno">  221</span>&#160;<span class="keywordtype">double</span> Ramp(<span class="keywordtype">double</span> u, <span class="keywordtype">double</span> tgt, <span class="keywordtype">double</span> inc){</div>
<div class="line"><a name="l00222"></a><span class="lineno">  222</span>&#160;    <span class="keywordtype">double</span> output;</div>
<div class="line"><a name="l00223"></a><span class="lineno">  223</span>&#160;    <span class="keywordflow">if</span> (abs(u - tgt) &lt; inc)</div>
<div class="line"><a name="l00224"></a><span class="lineno">  224</span>&#160;        output = tgt;</div>
<div class="line"><a name="l00225"></a><span class="lineno">  225</span>&#160;    <span class="keywordflow">else</span> <span class="keywordflow">if</span> (u &lt; tgt - inc)</div>
<div class="line"><a name="l00226"></a><span class="lineno">  226</span>&#160;        output = u + inc;</div>
<div class="line"><a name="l00227"></a><span class="lineno">  227</span>&#160;    <span class="keywordflow">else</span> <span class="keywordflow">if</span> (u &gt; tgt + inc)</div>
<div class="line"><a name="l00228"></a><span class="lineno">  228</span>&#160;        output = u - inc;</div>
<div class="line"><a name="l00229"></a><span class="lineno">  229</span>&#160;    <span class="keywordflow">else</span></div>
<div class="line"><a name="l00230"></a><span class="lineno">  230</span>&#160;        output = tgt;</div>
<div class="line"><a name="l00231"></a><span class="lineno">  231</span>&#160; </div>
<div class="line"><a name="l00232"></a><span class="lineno">  232</span>&#160;    <span class="keywordflow">return</span> output;</div>
<div class="line"><a name="l00233"></a><span class="lineno">  233</span>&#160;}</div>
<div class="line"><a name="l00234"></a><span class="lineno">  234</span>&#160; </div>
<div class="line"><a name="l00235"></a><span class="lineno">  235</span>&#160;<span class="keywordtype">void</span>  Limit(<span class="keywordtype">double</span> &amp;data, <span class="keywordtype">double</span> max, <span class="keywordtype">double</span> min){</div>
<div class="line"><a name="l00236"></a><span class="lineno">  236</span>&#160;    <span class="keywordflow">if</span> (data &gt; max)</div>
<div class="line"><a name="l00237"></a><span class="lineno">  237</span>&#160;        data = max;</div>
<div class="line"><a name="l00238"></a><span class="lineno">  238</span>&#160;    <span class="keywordflow">if</span> (data &lt; min)</div>
<div class="line"><a name="l00239"></a><span class="lineno">  239</span>&#160;        data = min;</div>
<div class="line"><a name="l00240"></a><span class="lineno">  240</span>&#160;}</div>
<div class="line"><a name="l00241"></a><span class="lineno">  241</span>&#160; </div>
<div class="line"><a name="l00242"></a><span class="lineno">  242</span>&#160; </div>
<div class="line"><a name="l00243"></a><span class="lineno">  243</span>&#160; </div>
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